Method of moments for a hidden serial-number range

Question

A firm's servers are labeled 1, 2, \dots, N for some unknown N. Orders get routed to a server chosen uniformly at random, and you record the labels of the servers that handled a large sample of your orders. The average label you observe is 60.5. Use the method of moments to estimate N.

Accepted Answer

Match the mean. A number drawn uniformly from 1 to N has mean sitting right in the middle: \mathbb{E}[X] = \frac{N + 1}{2}. Solve. Set that equal to the observed average of 60.5: \frac{N + 1}{2} = 60.5 \implies N + 1 = 121 \implies \hat{N} = 120. A caveat worth knowing. This is a famous setup (the "tank counting" problem). Method of moments gives a quick, reasonable answer, but it only uses the average and can occasionally return an estimate smaller than the largest label you actually saw, which is impossible. A sharper estimator leans on the sample maximum instead. Still, matching the mean i…

Method of moments for a hidden serial-number range

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